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Theorem alrimii 31819
Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
alrimii.1  |-  F/ y
ph
alrimii.2  |-  ( ph  ->  ps )
alrimii.3  |-  ( [. y  /  x ]. ch  <->  ps )
alrimii.4  |-  F/ y ch
Assertion
Ref Expression
alrimii  |-  ( ph  ->  A. x ch )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem alrimii
StepHypRef Expression
1 alrimii.1 . . 3  |-  F/ y
ph
2 alrimii.2 . . . 4  |-  ( ph  ->  ps )
3 alrimii.3 . . . 4  |-  ( [. y  /  x ]. ch  <->  ps )
42, 3sylibr 214 . . 3  |-  ( ph  ->  [. y  /  x ]. ch )
51, 4alrimi 1903 . 2  |-  ( ph  ->  A. y [. y  /  x ]. ch )
6 nfsbc1v 3299 . . 3  |-  F/ x [. y  /  x ]. ch
7 alrimii.4 . . 3  |-  F/ y ch
8 sbceq2a 3291 . . 3  |-  ( y  =  x  ->  ( [. y  /  x ]. ch  <->  ch ) )
96, 7, 8cbval 2050 . 2  |-  ( A. y [. y  /  x ]. ch  <->  A. x ch )
105, 9sylib 198 1  |-  ( ph  ->  A. x ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186   A.wal 1405   F/wnf 1639   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-sbc 3280
This theorem is referenced by: (None)
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